Theorem cycls

Description: The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 31-Jan-2021)

	Ref	Expression
Assertion	cycls	$$⊢\mathrm{Cycles}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Paths}\left(G\right)p\wedge p\left(0\right)=p\left(\left|f\right|\right)\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		biidd	$$⊢\left(\top \wedge g=G\right)\to \left(p\left(0\right)=p\left(\left|f\right|\right)\leftrightarrow p\left(0\right)=p\left(\left|f\right|\right)\right)$$
2		wksv	$$⊢\left\{⟨f,p⟩|f\mathrm{Walks}\left(G\right)p\right\}\in \mathrm{V}$$
3		pthiswlk	$$⊢f\mathrm{Paths}\left(G\right)p\to f\mathrm{Walks}\left(G\right)p$$
4	3	ssopab2i	$$⊢\left\{⟨f,p⟩|f\mathrm{Paths}\left(G\right)p\right\}\subseteq \left\{⟨f,p⟩|f\mathrm{Walks}\left(G\right)p\right\}$$
5	2 4	ssexi	$$⊢\left\{⟨f,p⟩|f\mathrm{Paths}\left(G\right)p\right\}\in \mathrm{V}$$
6	5	a1i	$$⊢\top \to \left\{⟨f,p⟩|f\mathrm{Paths}\left(G\right)p\right\}\in \mathrm{V}$$
7		df-cycls	$$⊢\mathrm{Cycles}=\left(g\in \mathrm{V}⟼\left\{⟨f,p⟩|\left(f\mathrm{Paths}\left(g\right)p\wedge p\left(0\right)=p\left(\left|f\right|\right)\right)\right\}\right)$$
8	1 6 7	fvmptopab	$$⊢\top \to \mathrm{Cycles}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Paths}\left(G\right)p\wedge p\left(0\right)=p\left(\left|f\right|\right)\right)\right\}$$
9	8	mptru	$$⊢\mathrm{Cycles}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Paths}\left(G\right)p\wedge p\left(0\right)=p\left(\left|f\right|\right)\right)\right\}$$